Given natural numbers $n$ and $k$, let $G_{n,k}$ denote the simple graph whose vertex set is $\{1,2\ldots ,n\}$ and there is an edge between $i$ and $j$ when $|i-j|\leq k$. I am interested in the independence number (size of the maximum independent set) of the graph $(G_{n,k})^{\Box d}$ (i.e. Cartesian product of $d$ copies of $G_{n.k}$). Are there any results known about such graphs? Even any results for small values of $d$ (e.g. $d=2,3$) would also be interesting to me.

Given natural numbers $n$ and $k$, let $G_{n,k}$ denote the simple graph whose vertex set is $\{1,2,\ldots ,n\}$ and there is an edge between $i$ and $j$ when $|i-j|\leq k$. I am interested in the independence number (size of the maximum independent set) of the graph $(G_{n,k})^{\Box d}$ (i.e. Cartesian product of $d$ copies of $G_{n.k}$). Are there any results known about such graphs? Even any results for small values of $d$ (e.g. $d=2,3$) would also be interesting to me.